Wand, M. Final algebra semantics and data type extensions. Journal of Computer and System Sciences 19, 1979, 27-44. 27  Home/Search   Document Not in Database   Summary   Related Articles   Check

....68Q60, 68Q65, 03B70 (AMS 91) F.3.1, F.3.2, D.1. 5 (CR 91) 1 Introduction This paper is part of a recent research line of applying coalgebraic and coinductive notions and techniques in the formalisation of object oriented concepts, see [25, 13, 10, 12, 14, 5, 6] building on earlier work [29, 2, 15]. Coalgebras consist of a state space together with a transition function and can be used to describe various kinds of dynamical systems, including automata, transition systems and hybrid systems, see e.g. 27, 20, 11] A coalgebraic specification (as developed in [13] formally captures several ....

M. Wand. Final algebra semantics and data type extension. Journ. Comp. Syst. Sci, 19:27--44, 1979. 18

....follows from them, and nothing else. This loose semantic interpretation guarantees that formulas in the theory follow only from the presence of assertions in the trait never from their absence. This is in contrast to algebraic speci cation languages based on initial algebras [6] or nal algebras [14]. Using the loose interpretation ensures that all theorems proved about an incomplete speci cation remain valid when it is extended. Each trait should be consistent: it must not de ne a theory containing the formula false. Consistency is often dicult to prove and is undecidable in general. ....

M. Wand. \Final algebra semantics and data type extensions." Journal of Computer and System Sciences, August 1979. 59

.... states, identifying two such machines iff their behaviour is equivalent [19, 46] This gives a semantics for abstract objects which generalises that of abstract data types, as in the semantics of FOOPS [23] An early attempt to handle state appears in Guttag s thesis [31] later formalised by Wand [60] using final algebras. However, approaches which admit a class of models, e.g. those models that are behaviourally equivalent, seem more satisfactory. Section 3.2 gives another such approach, based on the satisfaction of equations up to observability. The existence of applications using many ....

Mitchell Wand. Final algebra semantics and data type extension. Journal of Computer and System Sciences, 19:27--44, 1979.

....whenever u; v 2 T (F ) and R j= u = v. 5 Observational semantics The notion of observation technique (see e.g. 4] has been introduced as a mean for describing what is observed in a given algebra. Various observation techniques have been proposed in the literature: observations based on sorts [36,31,28,16], on operators [1] on terms [34,15,5] or on formulas [33,34,22] An observational speci cation is then obtained by adding an observation technique to a standard algebraic speci cation. Our observation technique is based on sorts but can easily be extended to operators. Our observational ....

M. Wand. Final algebra semantics and data type extensions. Journal of Computer and System Sciences, 19:27-44, 1979. 29

....follows from them, and nothing else. This loose semantic interpretation guarantees that formulas in the theory follow only from the presence of assertions in the trait never from their absence. This is in contrast to algebraic speci cation languages based on initial algebras [6] or nal algebras [14]. Using the loose interpretation ensures that all theorems proved about an incomplete speci cation remain valid when it is extended. Each trait should be consistent: it must not de ne a theory containing the formula false. Consistency is often dicult to prove and is undecidable in general. ....

M. Wand. \Final algebra semantics and data type extensions." Journal of Computer and System Sciences, August 1979. 56

....abstract semantics of programming languages [Mil77] The main purpose of this paper is to formalize and study abstract theories and the corresponding characteristic models which we call the optimal algebras. The best existing formalization of abstract theories is found in final algebra semantics [Wan77, Kam83]. However, this formalization has a serious flaw it is only applicable to specifications that are complete in some sense. For instance, the specification given above is incomplete because it does not specify the result of expressions of the form member(x,empty) Final algebra semantics is not ....

....observable sort; i.e. it guarantees that the carriers of observable sorts will be exactly those in I base . A specification (base, ext) is said to be sufficiently complete iff I ext is standard [GH78] The main theorem of final algebra semantics is: 4 THREE FUNDAMENTAL MODALITIES 6 Theorem 1 ([Wan77]) For every sufficiently complete specification (base; ext) the category Rext has a final object. 4 Three Fundamental Modalities We show in this section that if the ideas of necessity and possibility are applied as modal operators to define a modal equational logic, then they give rise to ....

[Article contains additional citation context not shown here]

M. Wand. Final algebra semantics and data type extensions. Journal of Computer and System Sciences, 19(1):27--44, 1977.

....meaningful and is hence left out. We do not know of previous work which discusses this style of object abstraction in a strongly typed framework. Abstraction functions from type implementations to type de nitions were rst introduced in [9] and were further developed in algebraic data type theory [7, 20]. These works concentrate on correctness proofs of a data type s implementation with respect to its speci cation; the case where the interface (i.e speci cation) and the implementation are parts of the same program is not considered. Programming languages such as CLU [13] which incorporate ....

Wand M., Final algebra semantics and data type extensions. J. Comput. Syst. Sci. 19(1), pages 27-44, 1979.

....coinduction eliminates the awkwardness of context induction. Reichel s seminal work on behavioral satisfaction was in part motivated by an insight on how to unify initial and nal semantics [85] Behavioral and nal semantics were perhaps rst advocated by Montanari et al. 25] though Wand [99] also made an early contribution. Finality is also used for treating states in [85,48,78,75] among many other places, including the present paper; there is some elegant more recent by work by Reichel on co algebraic semantics for the object paradigm [87] Some sophisticated results on ....

Mitchell Wand. Final algebra semantics and data type extension. Journal of Computer and System Sciences, 19:27-44, 1979.

....common observable context. Consequently, these two values cannot be compared. However, according to our Indistinguishability Assumption, we do not consider that two elements can either be indistinguishable, distinguishable or incomparable. Our point of view is close to final semantics ( 3] 13] [20]) we consider indistinguishable these pairs of elements, for which we do not observe the contrary. This is stated in the undermentioned definition. For a while assume already defined the notion of observable contexts w.r.t. a set W of observable terms. Definition (comparator, version 1) We call ....

Wand M. Final Algebra Semantics and Data Type Extension Journal of Computer and System Sciences, Vol 19, 27-44 (1979).

.... items are equivalent i they can be proved equal from the axioms) GTWW75, EM85] In order to guarantee executability and the existence of initial models, one usually restricts the axioms to conditional equations 1 Several alternative approaches have been proposed (e.g. nal algebra semantics [Wan81] but are of less practical relevance. 2 We give precise de nitions in section 2. or even equations only. Some examples of speci cation languages following the initial approach are Clear, OBJ, and ACT ONE. In the loose approach the semantics of a speci cation is de ned to be the set of all ....

M. Wand. Final algebra semantics and data type extensions. J. Comput. System Sci., 19:27-44, 1981.

....convergent if the terms u and v are joinable whenever u; v 2 T (F ) and R j= u = v. 4 Observational semantics The notion of observation technique have been introduced as a mean for describing what is observed in a given algebra. Various techniques have been proposed: observations based on sorts [Wan79, Rei84, NO87, Hen91], operators[BB91] terms [ST88, Hen89, BBK92] or formula [ST85, ST88, Kna91] The observation technique we use in our method is based on sorts 1 . The semantic we choose is based on a relaxing of the satisfaction relation. The notion of context is fundamental in all approaches based on such ....

M. Wand. Final algebra semantics and data type extensions. Journal of Computer and System Sciences, 19:27--44, 1979.

....may consider that the enrichment operation adds expressive power to our specification language (see [2] 4. Other constructors that have been used in the literature to define the semantics of enrichments are associated to other kinds of (monomorphic) semantics, such as final semantics (see e.g. [24,42]) or behavioural semantics (see e.g. 36, 33] 3.2 Union This is the other most basic operation for building specifications incrementally. The idea is that we build a new specification by putting together two smaller ones, which may share some subspecification. For instance, the above ....

M. Wand, Final Algebra Semantics and Data Type Extensions, Journ. of Comp. and System Sciences 19 (1979) pp. 27-44.

.... with a liveness predicate expressing divergence (see [Gor95] 10.3 Final semantics and bisimilarity Final semantics was introduced for modelling permutative types such as finite sets, finite bags (multisets) and functions with a finite domain (stores, arrays, indexed lists) see, e.g. GGM76,Wan79,Kam83] These types are still constructor based, but constructor equations are needed to axiomatize data equality. Hence specifications of permutative types are complete, but 10 Proof in Flat Specifications 19 not consistent. From a model theoretic viewpoint, initial semantics is sufficient for ....

M. Wand. Final algebra semantics and data type extensions. Journal of Computer and System Sciences, 19:27--44, 1979.

....functors from models to domains , which generalizes equational logic to cover Horn clause logic and constraint logic programming. Initiality properties are used to prove a Herbrand theorem at a very high level of abstraction and generality. Final semantics In the case of final semantics [376, 933], the interesting models are considered to be the terminal objects of a certain sub category of the category of all the models. Usually such subcategory is chosen in such a way that final algebra semantics represents the mathematical semantics of a specification by which the identity of a data ....

M. Wand. Final algebra semantics and data type extensions. Journal of Computer and System Sciences, 19:27--44, 1979.

.... model specification [LB 77] or specification by example [Sad 84] In general, specifications under the usual algebraic approaches are not abstract enough; it is either difficult, as in Clear [BG 80] or impossible, as in the initial algebra approach of [ADJ 76] and the final algebra approach of [Wand 79] to specify sets of natural numbers in such a way that both A and B above are models of the specification. ASL provides a behavioural abstraction operation which when applied to a specification SP relaxes interpretation to all those algebras which are behaviourally equivalent to a model of SP . ....

Wand, M. Final algebra semantics and data type extensions. J. Computer and System Sciences 19, pp. 27-44.

....from them, and nothing else. This loose semantic interpretation guarantees that formulas in the theory follow only from the presence of assertions in the trait never from their absence. This is in contrast to algebraic specification languages based on initial algebras [5] or final algebras [13]. Using the loose interpretation ensures that all theorems proved about an incomplete specification remain valid when it is extended. Each trait should be consistent: it must not define a theory containing the formula false. Consistency is often difficult to prove and is undecidable in general. ....

M. Wand. "Final algebra semantics and data type extensions." Journal of Computer and System Sciences, August 1979.

....equational specification and universal algebra are the bases of abstract data types specification and semantics, we 3. A Formal Framework of Hardware Specification 5 do not attempt to associate our hardware specification with any of the particular abstract data type semantics proposed in [2, 3, 4, 5, 6, 7]. Some of the definitions were inspired by [2] More detailed discussion on the specifying hardware architecture in an algebraic framework can be found in [8, 9] 3.1 Specifying Hardware Architecture and System Let S be a set of sorts. An S sorted signature Sigma is an S S sorted family ....

Wand, M. Final algebra semantics and data type extensions. Journal of Computing System Science 19 (1979), 27--44.

....These paradigms collectively span the permitted behaviors, and one must choose a paradigm and then make a comparison with the candidate implementation. We first motivate the need for a collection of paradigms, and then discuss techniques for making the comparison. The initial [7] and final [18] algebra approaches, are ways to choose a paradigm that is a unique (up to isomorphism) However, the application of such an approach to incomplete specifications that are hierarchical over some primitive types is misguided and does not work [5] 20] As an example, imagine an ADT Set with an ....

Mitchell Wand, Final Algebra Semantics and Data Type Extensions, Journal of Computer and System Sciences 19 (1979), no. 1, 27--44.

.... in examples (which are fully formalised and verified in pvs) 1 Introduction This paper is part of a recent research line of applying coalgebraic and coinductive notions and techniques in the formalisation of object oriented concepts, see [26, 14, 11, 13, 15, 5, 6] building on earlier work [30, 2, 17]. Coalgebras consist of a state space together with a transition function and can be used to describe various kinds of dynamical systems, including automata, transition systems and hybrid systems, see e.g. 28, 22, 12] or [16] for an introduction to the theory of coalgebras) A coalgebraic ....

M. Wand. Final algebra semantics and data type extension. Journ. Comp. Syst. Sci, 19:27--44, 1979.

....that the formulas in the theory follow only from the presence of assertions in the trait never from their absence. This is in contrast to algebraic specification languages based on initial or final algebras [Ehrig and Mahr 1985; Goguen, Thatcher, and Wagner 1978; Sanella and Tarlecki 1987; Wand 1979] Our interpretation is essential 1 LSL has a very simple precedence scheme for operators: postfix operators consisting of a period followed by an identifier bind most tightly. Other user defined operators and the built in Boolean negation operator ( bind more tightly than the built in in ....

M. Wand, "Final Algebra Semantics and Data Type Extensions," Journal of Computer and System Sciences, vol. 19, no. 1, pp. 27--44, 1979.

....common observable context. Consequently, these two values cannot be compared. However, according to our Indistinguishability Assumption, we do not consider that two elements can either be indistinguishable, distinguishable or incomparable. Our point of view is close to final semantics ( 3] 11] [18]) we consider indistinguishable these pairs of elements, for which we do not observe the contrary. This is stated in the definition below (for a while assume already defined the notion of observable contexts) Definition (comparator, version 1) We call W comparator (or shortly comparator) of ....

Wand M. Final Algebra Semantics and Data Type Extension Journal of Computer and System Sciences, Vol 19, 27-44 (1979)

No context found.

Wand, M. Final algebra semantics and data type extensions. Journal of Computer and System Sciences 19, 1979, 27-44. 27

No context found.

Mitchtc Wand. Final algebra semantics and data type extension. Journal of Computer and System Sciences, 19:27--44, 1979.

No context found.

M. Wand, Final Algebra Semantics and Data Type Extensions, J. Computer and System Sciences 19 (1979) 27-44

No context found.

M. Wand. Final algebra semantics and data type extension. Journ. Comp. Syst. Sci, 19:27--44, 1979. 18

First 50 documents

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC